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arXiv:2610.05393

Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle

Jiro Akahori, Riki Kitano, Norio Konno, Iwao Sato

math.PRmath-phquant-phSubmitted Oct 4, 2026• Announced Oct 6, 2026
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Abstract

We establish a local Konno-Sato correspondence for reflected random and Szegedy walks on a finite path graph. A generalized reflection principle, using explicit involutions rather than point reflections in the asymmetric case, yields exact zeta factorizations into a boundary factor and local factors indexed by the infinite dihedral group. After symmetrization, the classical and quantum local factors correspond for each group element. The identity contribution determines the normalized thermodynamic limits. In the symmetric case, diffusive and ballistic scalings recover the Gaussian Poisson summation formula and the Dirac comb identity, respectively. Under a double scaling of the asymmetry and spectral parameter, the quantum factorization yields a massive Dirac-type limit: nonidentity even words produce the periodic-orbit factor, the identity produces the free-field factor, and odd words together with the boundary produce a finite boundary correction.

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We establish a local Konno-Sato correspondence for reflected random and Szegedy walks on a finite path graph. A generalized reflection principle, using explicit involutions rather than point reflections in the asymmetric case, yields exact zeta factorizations into a boundary factor and local factors indexed by the infinite dihedral group. After symmetrization, the classical and quantum local factors correspond for each group element. The identity contribution determines the normalized thermodynamic limits. In the symmetric case, diffusive and ballistic scalings recover the Gaussian Poisson summation formula and the Dirac comb identity, respectively. Under a double scaling of the asymmetry and spectral parameter, the quantum factorization yields a massive Dirac-type limit: nonidentity even words produce the periodic-orbit factor, the identity produces the free-field factor, and odd words together with the boundary produce a finite boundary correction.
In this paper, following the recent paper on Walk/Zeta Correspondence by the first author and his coworkers, we compute the zeta function for the three- and four-state quantum walk and correlated random walk, and the multi-state random walk on the one-dimensional torus by using the Fourier analysis. We deal with also the four-state quantum walk and correlated random walk on the two-dimensional torus. In addition, we introduce a new class of models determined by the generalized Grover matrix bridging the gap between the Grover matrix and the positive-support of the Grover matrix. Finally, we give a generalized version of the Konno-Sato theorem for the new class. As a corollary, we calculate the zeta function for the generalized Grover matrix on the d-dimensional torus.
We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.
Quantum walks are discrete-time evolutions of single particles on lattices that transfer the concept of classical random walks to quantum theory. Topological insulators are objects that promise symmetry-protected edge-states for transportation along the edge. Together, both subjects lay the foundation to a description of new materials that may be used e.g. for quantum computation or transport. In this thesis, we study topological phases in quantum walks. Given a representation of the discrete symmetry groups of the tenfold way, we develop a topological classification which distinguishes three symmetry indices whose values lie in the group of integers, the group of two elements, or the trivial group, depending on the symmetry type under consideration. The classification is applicable to quantum walks that are gapped at the symmetry-protected points. All symmetry indices are proven to be stable under norm-continuous perturbations, but only two of the three indices are invariant under compact non-continuous perturbations. These two indices describe the asymptotic behaviour far to the left and far to the right, respectively. The third index reflects whether a compact perturbation can be performed in arbitrarily small steps without violating the symmetries. Given two walks in the same phase (i.e. the symmetry indices coincide), we show that there is a norm-continuous path of walks in that phase that connects them. This renders the set of invariants we introduced complete. Our theory covers translation invariant bulks as well, where we prove that the third index vanishes and the left- and right indices add up to zero. Hence these systems are fully classified by the right index alone. Joining two bulks in different phases (one left, one right), we show that eigenvalues emerge in the gap (bulk-boundary correspondence), whose eigenfunctions decay exponentially away from the boundary. Since our theory does not demand translation invariance at all, these composed systems are still described by our general classification. Restricting to the translation invariant case, we express the symmetry index as a winding number of a loop in momentum space. Within this restricted class, we prove that our classification is complete as well.
In this paper, we focus on discrete-time random walks (DTRWs) on the path graph with reflecting walls. Also we consider the corresponding continuous-time random walks (CTRWs), discrete-time quantum walks (DTQWs) and continuous-time quantum walks (CTQWs). We give formulas for the return probabilities of these four walks (DTRWs, CTRWs, DTQWs, CTQWs) at the boundary point of the path graph and discuss their relationships. Furthermore, by considering the Ehrenfest model, we derive the scaling limit theorems for the return probabilities of both the CTRW and the CTQW.
We consider general quantum random walks in a d -dimensional half-space. We first obtain a path integral formula for general quantum random walks in a d -dimensional space. Our path integral formula is valid for general quantum random walks on Cayley graphs as well. Then the path integral formula is applied to obtain the scaling limit of the exit distribution, the expectation of exit time and the asymptotic behaviour of the exit probabilities, for general quantum random walks in a half-space under some conditions on amplitude functions. The conditions are shown to be satisfied by both the Hadamard and Grover quantum random walks in two-dimensional half-spaces. For the two-dimensional case, we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero, i.e. the natural magnitude of the hitting position is of order O (1) if the lattice spacing is set to be 1/ n . We also show that the rate of convergence of the total hitting probability has lower bound n −2 and upper bound n −2+ for any > 0. For a quantum random walk with a fixed starting point, we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk. In both one and two dimensions, given the event of a hit, the conditional expectation of hitting times is finite, in contrast to being infinite for the classical case. In the one-dimensional case, we also obtain an exact order of the probability distribution of the hitting time at 0.